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Heap (data Structure)
In computer science, a heap is a specialized tree-based data structure which is essentially an almost complete tree that satisfies the heap property: in a ''max heap'', for any given node C, if P is a parent node of C, then the ''key'' (the ''value'') of P is greater than or equal to the key of C. In a ''min heap'', the key of P is less than or equal to the key of C. The node at the "top" of the heap (with no parents) is called the ''root'' node. The heap is one maximally efficient implementation of an abstract data type called a priority queue, and in fact, priority queues are often referred to as "heaps", regardless of how they may be implemented. In a heap, the highest (or lowest) priority element is always stored at the root. However, a heap is not a sorted structure; it can be regarded as being partially ordered. A heap is a useful data structure when it is necessary to repeatedly remove the object with the highest (or lowest) priority, or when insertions need to be interspe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Search Tree
In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree. The time complexity of operations on the binary search tree is directly proportional to the height of the tree. Binary search trees allow binary search for fast lookup, addition, and removal of data items. Since the nodes in a BST are laid out so that each comparison skips about half of the remaining tree, the lookup performance is proportional to that of binary logarithm. BSTs were devised in the 1960s for the problem of efficient storage of labeled data and are attributed to Conway Berners-Lee and David Wheeler. The performance of a binary search tree is dependent on the order of insertion of the nodes into the tree since arbitrary insertions may lead to degeneracy; several variations of the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brodal Queue
In computer science, the Brodal queue is a heap/priority queue structure with very low worst case In computer science, best, worst, and average cases of a given algorithm express what the resource usage is ''at least'', ''at most'' and ''on average'', respectively. Usually the resource being considered is running time, i.e. time complexity, b ... time bounds: O(1) for insertion, find-minimum, meld (merge two queues) and decrease-key and O(\mathrm(n)) for delete-minimum and general deletion. They are the first heap variant to achieve these bounds without resorting to amortization of operational costs. Brodal queues are named after their inventor Gerth Stølting Brodal.Gerth Stølting Brodal (1996). Worst-case efficient priority queues. Proc. 7th ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58 While having better asymptotic bounds than other priority queue structures, they are, in the words of Brodal himself, "quite complicated" and " otapplicable in practice." Brodal and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Heap
In computer science, a binomial heap is a data structure that acts as a priority queue but also allows pairs of heaps to be merged. It is important as an implementation of the mergeable heap abstract data type (also called meldable heap), which is a priority queue supporting merge operation. It is implemented as a heap similar to a binary heap but using a special tree structure that is different from the complete binary trees used by binary heaps. Binomial heaps were invented in 1978 by Jean Vuillemin. Binomial heap A binomial heap is implemented as a set of binomial trees (compare with a binary heap, which has a shape of a single binary tree), which are defined recursively as follows: * A binomial tree of order 0 is a single node * A binomial tree of order k has a root node whose children are roots of binomial trees of orders k-1, k-2, ..., 2, 1, 0 (in this order). A binomial tree of order k has 2^k nodes, and height k. The name comes from the shape: a binomial tree o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Heap
A binary heap is a heap data structure that takes the form of a binary tree. Binary heaps are a common way of implementing priority queues. The binary heap was introduced by J. W. J. Williams in 1964, as a data structure for heapsort. A binary heap is defined as a binary tree with two additional constraints: *Shape property: a binary heap is a ''complete binary tree''; that is, all levels of the tree, except possibly the last one (deepest) are fully filled, and, if the last level of the tree is not complete, the nodes of that level are filled from left to right. *Heap property: the key stored in each node is either greater than or equal to (≥) or less than or equal to (≤) the keys in the node's children, according to some total order. Heaps where the parent key is greater than or equal to (≥) the child keys are called ''max-heaps''; those where it is less than or equal to (≤) are called ''min-heaps''. Efficient (logarithmic time) algorithms are known for the two op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beap
A beap, or bi-parental heap, is a data structure where a node usually has two parents (unless it is the first or last on a level) and two children (unless it is on the last level). Unlike a heap, a beap allows sublinear search. The beap was introduced by Ian Munro and Hendra Suwanda. A related data structure is the Young tableau. Performance The height of the structure is approximately \sqrt. Also, assuming the last level is full, the number of elements on that level is also \sqrt. In fact, because of these properties all basic operations (insert, remove, find) run in O(\sqrt) time on average. Find operations in the heap can be O(n) in the worst case. Removal and insertion of new elements involves propagation of elements up or down (much like in a heap) in order to restore the beap invariant. An additional perk is that beap provides constant time access to the smallest element and O(\sqrt) time for the maximum element. Actually, a O(\sqrt) find operation can be implemented i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
B-heap
A B-heap is a binary heap implemented to keep subtrees in a single page. This reduces the number of pages accessed by up to a factor of ten for big heaps when using virtual memory, compared with the traditional implementation. The traditional mapping of elements to locations in an array puts almost every level in a different page. There are other heap variants which are efficient in computers using virtual memory or caches, such as cache-oblivious algorithms, k-heaps, and van Emde Boas layouts. Motivation Traditionally, binary trees are laid out in consecutive memory according to a n -> rule, meaning that if a node is at position n, its left and right child are taken to be at positions 2n and 2n+1 in the array. The root is at position 1. A common operation on binary trees is the vertical traversal; stepping down through the levels of a tree in order to arrive at a searched node. However, because of the way memory is organized on modern computers into pages in virtual memory, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2–3 Heap
In computer science, a 2–3 heap is a data structure, a variation on the heap, designed by Tadao Takaoka in 1999. The structure is similar to the Fibonacci heap, and borrows from the 2–3 tree. Time costs for some common heap operations are: * ''Delete-min'' takes O(\log(n)) amortized time In computer science, amortized analysis is a method for analyzing a given algorithm's complexity, or how much of a resource, especially time or memory, it takes to execute. The motivation for amortized analysis is that looking at the worst-case .... * ''Decrease-key'' takes constant amortized time. * ''Insertion'' takes constant amortized time. Polynomial of trees A linear tree of size r is a sequential path of r nodes with the first node as a root of the tree and it is represented by a bold \mathbf (e.g. \mathbf is a linear tree of a single node). Product P = ST of two trees S and T, is a new tree with every node of S is replaced by a copy of T and for each edge of S we connect the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling's Approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. One way of stating the approximation involves the logarithm of the factorial: \ln(n!) = n\ln n - n +O(\ln n), where the big O notation means that, for all sufficiently large values of n, the difference between \ln(n!) and n\ln n-n will be at most proportional to the logarithm. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to use instead the binary logarithm, giving the equivalent form \log_2 (n!) = n\log_2 n - n\log_2 e +O(\log_2 n). The error term in either base can be expressed more precisely as \tfrac12\log(2\pi n)+O(\tfrac1n), corresponding to an approximate formula for the factorial itself, n! \sim \sqrt\left(\frac\righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heapsort
In computer science, heapsort is a comparison-based sorting algorithm. Heapsort can be thought of as an improved selection sort: like selection sort, heapsort divides its input into a sorted and an unsorted region, and it iteratively shrinks the unsorted region by extracting the largest element from it and inserting it into the sorted region. Unlike selection sort, heapsort does not waste time with a linear-time scan of the unsorted region; rather, heap sort maintains the unsorted region in a heap data structure to more quickly find the largest element in each step. Although somewhat slower in practice on most machines than a well-implemented quicksort, it has the advantage of a more favorable worst-case runtime (and as such is used by Introsort as a fallback should it detect that quicksort is becoming degenerate). Heapsort is an in-place algorithm, but it is not a stable sort. Heapsort was invented by J. W. J. Williams in 1964. This was also the birth of the heap, presented ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implicit Data Structure
In computer science, an implicit data structure or space-efficient data structure is a data structure that stores very little information other than the main or required data: a data structure that requires low overhead. They are called "implicit" because the position of the elements carries meaning and relationship between elements; this is contrasted with the use of pointers to give an ''explicit'' relationship between elements. Definitions of "low overhead" vary, but generally means constant overhead; in big O notation, ''O''(1) overhead. A less restrictive definition is a succinct data structure, which allows greater overhead. Definition An implicit data structure is one with constant space overhead (above the information-theoretic lower bound). Historically, defined an implicit data structure (and algorithms acting on one) as one "in which structural information is implicit in the way data are stored, rather than explicit in pointers." They are somewhat vague in the defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
